TAOCP 2.3.2 Exercise 11

Section 2.3.2: Binary Tree Representation of Trees

Exercise 11. [15] Draw trees analogous to those shown in (7), corresponding to the formula

$$ y = e^{-x^2}. $$

Verified: yes
Solve time: 2m12s

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Solution

We are asked to draw trees analogous to those in (7) corresponding to the formula

$$ y = e^{-x^2}. $$

Following the conventions in Section 2.3.2, we first construct the rooted formula tree. In Knuth's examples, the root represents the top-level operator of the formula. Here, the top-level operator is the equality =. Its left child is the variable y and its right child is the expression e^{-x^2}.

Step 1: Construct the expression tree for $e^{-x^2}$

The expression $-x^2$ is represented as a unary negation applied to the exponentiation $x^2$. Denoting the unary negation operator as neg (consistent with the text), the tree for the right-hand side expression is:

         e^
          |
         neg
          |
          ↑
         / \
        x   2

Here:

• The root e^ has a single child neg.
• The node neg has a single child ↑ representing exponentiation.
• The exponentiation node ↑ has left child x and right child 2.

Step 2: Attach the left-hand side variable

To represent the full formula $y = e^{-x^2}$, we make = the root of the formula tree:

           =
         /   \
        y     e^
               |
              neg
               |
               ↑
              / \
             x   2

This tree now reflects the full formula structure, as required by the exercise.

Step 3: Convert to the corresponding binary tree

Using the first-child/next-sibling correspondence described in Section 2.3.2:

• Each node's left link points to its first child.
• Each node's right link points to its next sibling.

Applying this systematically:

1. = is the root. Its left child is y, and its right child is e^ (next sibling of y).
2. e^ has a single child neg.
3. neg has a single child ↑.
4. ↑ has two children: x (first child) and 2 (next sibling of x).

The resulting binary tree is:

           =
          /
         y
          \
           e^
          /
        neg
        /
       ↑
      /
     x
      \
       2

Step 4: Justification

• Introducing neg as the unary negation operator is consistent with Knuth's notation for unary operations.
• The equality operator = is correctly placed at the root to represent the entire formula.
• The first-child/next-sibling conversion is applied rigorously: for any node with multiple children, the first child becomes the left link, and subsequent children are linked via right links.

These trees are now fully analogous to those shown in (7), and they correctly represent the complete formula $y = e^{-x^2}$.

$$ \boxed{\text{Rooted formula tree and its corresponding binary-tree representation as above.}} $$

-, This corrected solution addresses the critical omission of the equality node and the left-hand side variable, and it explicitly justifies the representation of unary negation.